Method for quantum annealing computation

ABSTRACT

A method for performing quantum annealing computation for solving a discrete optimization problem includes the steps of: (a) identifying an operation for transferring an energy state of one of two physical systems (u, d) having the same quantum state to a low energy state and transferring an energy state of the other of the two physical systems to a high energy state; (b) constructing a network structure among a plurality of physical systems that indicates an order of application of the operation of the step (a) on two physical systems among the plurality of physical systems; and (c) obtaining a physical system having a minimum energy state in the plurality of physical systems by applying the operation of the step (a) to the plurality of physical systems according to the order indicated in the network structure of step (b).

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates to a method for quantum annealingcomputation, and more particularly, to a method for performing quantumannealing computation to solve a discrete optimization problem using aquantum annealing computer.

Description of the Related Art

A quantum annealing computer is known as a quantum computer that solvesa discrete optimization problem using a quantum effect. The quantumannealing computer is a computer for solving a problem (hereinafterreferred to as a “discrete optimization problem”) of obtaining acombination of discrete variables that minimizes an objective function(corresponding to optimization), which frequently appears in operationresearch or the like.

A physical system is designed in which a discrete variable is a physicalstate and a value of an objective function with respect to the discretevariable becomes the energy of the state. That is, energy is a functionof the state. Thus, when the physical system can be transferred to alowest energy state by any method, an optimum solution can be obtainedby measuring the state. Quantum annealing using a quantum mechanicaleffect is known as a mechanism for transferring the physical system tothe lowest energy state. It is known that the quantum annealing maysolve a problem more efficiently than those that do not use the quantummechanical effect.

In the quantum annealing, correspondence between the energy and thestate is gradually changed in time by controlling an external field(potential) acting on a system. An initial potential having aself-evident lowest energy state is selected. The quantum annealing isdesigned to change the potential and to finally achieve a relationshipbetween the energy and the state corresponding to an objective functionfor which the optimum solution is actually investigated. When theinitial state of the system is prepared to be the lowest energy statedetermined by the initial potential and then the potential is changedsufficiently slowly, the state traces the lowest energy state determinedby the potential at each moment, which is known as a result of quantummechanics. In this manner, it is possible to obtain a state thatminimizes the objective function to be finally examined.

A fact that the state traces the lowest energy state when the timechange in the potential is slow is generally known as an adiabatictheorem in quantum mechanics, and the computation method is also knownas adiabatic quantum computation. As an example, in JP-A-2009-524857,disclosed is a method for the adiabatic quantum computation using aquantum system including a plurality of superconducting qubits. In theadiabatic quantum computation, a quantum mechanical state (a state thatcannot be expressed by ordinary classical physics) appears in a statethat appears in a computation process. Therefore, the computation methodmay solve the problem more efficiently than an ordinary computer usingclassical physics as an implementation principle.

SUMMARY OF THE INVENTION

In conventional quantum annealing computation shown in JP-A-2009-524857,a time change in potential must be slow so that an adiabatic theorem canbe established. For this reason, the computation cannot be acceleratedbeyond the limit. In particular, when a size of a problem to be solvedis big, there exists a problem (bottleneck) that an expectation value ofcomputation time increases exponentially with respect to a size of theproblem.

An object of the present invention is not only to propose a methoddifferent from the conventional method using the adiabatic theorem, as amechanism for guiding a physical state to the lowest energy state, butalso to avoid the above-described bottleneck problem.

The present invention provides a method for performing quantum annealingcomputation to solve a discrete optimization problem. The methodincluding the following (ii) as a basic idea includes:

-   -   (i) defining an energy function (problem Hamiltonian H) so that        a solution of a problem becomes a lowest energy state;    -   (ii) repeatedly applying a method for transferring an energy        state of one of two physical systems having the same quantum        state to a low energy state and transferring an energy state of        the other of the two physical systems to a high energy state, to        the two physical systems among a plurality of systems; and    -   (iii) determining the lowest energy state which is the solution        to the problem.

A method of the present invention for performing quantum annealingcomputation to solve a discrete optimization problem using a quantumannealing computer, the method including the step of;

-   -   (a) identifying an operation for transferring an energy state of        one of two physical systems (u, d) having the same quantum state        to a low energy state and transferring an energy state of the        other of the two physical systems to a high energy state;    -   (b) constructing a network structure among a plurality of        physical systems that indicates an order of application of the        operation of the step (a) on two physical systems among the        plurality of physical systems; and    -   (c) obtaining a physical system having a minimum energy state in        the plurality of physical systems by applying the operation of        the step (a) according to the order indicated in the network        structure of step (b).

As an example, when the method of the present invention is applied to asearch problem of a search space 2^(n), the number of computation stepscan be suppressed to a highly polynomial (second order (n²)) increasewith respect to n. Further, when the conventional method for quantumannealing computation and a conventional method for gate type quantumcomputation are applied, the number of computation steps increasesexponentially (2^(n/2)) with respect to n, whereas according to themethod of the present invention, the number of computation steps can besignificantly reduced. Further, when the method of the present inventionis used to solve the problem, the method of the present invention allowssolving the problem by the number of computation steps and the requirednumber of quantum bits with a highly polynomial size (second order (n²))with respect to n. As a result, an increase in computation timeaccording to the size of the problem can be suppressed.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram illustrating an outline of an example of acomputation method including a main part of an algorithm of a method ofthe present invention;

FIG. 2 is a diagram illustrating a flow of one embodiment of the methodof the present invention;

FIG. 3 is a conceptual diagram of an operation (step S10 in FIG. 2) fortransferring an energy state of one embodiment of the method of thepresent invention;

FIG. 4 is a schematic diagram of the conceptual diagram in FIG. 3;

FIG. 5 is a diagram illustrating step S20 of obtaining a set of oneembodiment of the method of the present invention and step S30 ofobtaining a physical system having a minimum energy state among aplurality of physical systems;

FIG. 6 is a diagram illustrating the time evolution of an expectationvalue of energy when one embodiment of the method of the presentinvention is performed (FIG. 5);

FIG. 7 is a diagram illustrating an implementation example(configuration example) in which the method of the present invention isapplied to a superconducting circuit (CJJ-rf-SQUID) using Josephsonjunction;

FIG. 8 is a diagram illustrating a control sequence of the embodiment ofFIG. 7;

FIG. 9 is a diagram illustrating n dependency of a temporal behavior ofa correct solution probability P(t) of a search problem by a nonlinearequation;

FIG. 10 is a diagram illustrating n dependency of time t* when thecorrect solution probability P(t) becomes P(t)≥0.9; and

FIG. 11 is a diagram illustrating m dependency of step* obtained bysubtracting 1 from step where Ω(step) becomes an empty set and thenumber of quantum operations C(m).

DESCRIPTION OF THE PREFERRED EMBODIMENTS

An embodiment of the present invention (a method for performing quantumannealing computation to solve a discrete optimization problem using aquantum annealing computer) will be described with reference to theaccompanying drawings. FIG. 1 is a diagram illustrating an outline of anexample of a computation method including a main part of an algorithm ofthe method. In an outline of the computation method in FIG. 1, first, anenergy function (problem Hamiltonian H) is defined so that a solution ofa problem becomes a lowest energy state (S1). Next, a method fortransferring an energy state of one of two physical systems having thesame quantum state to a low energy state and transferring an energystate of the other of the two physical systems to a high energy state isrepeatedly applied the two physical systems among the plurality ofphysical systems (S2). As a result, the lowest energy state whose stateis the solution of the problem can be obtained (S3).

FIG. 2 is a diagram illustrating a flow of one embodiment of the methodfor performing the quantum annealing computation of the presentinvention. As step S10, an operation for transferring an energy state ofone of two physical systems (u, d) having the same quantum state to alow energy state and transferring an energy state of the other of thetwo physical systems (u, d) to a high energy state is identified. Asstep S20, among the plurality of physical systems, each state value ofthe plurality of physical systems in each step is set under apredetermined rule while sequentially transferring steps from step 0which will be described later, and a set is obtained by collecting apair of physical systems having the same state value until reaching astep in which a pair of physical systems having the same state value iseliminated. In other words, in step S20, a network structure (refer toFIG. 5 which will be described later) among the plurality of physicalsystems that indicates an order of application of the operation of stepS10 on two physical systems among the plurality of physical systems isconstructed. In step S30, a physical system having a minimum energystate in the plurality of physical systems is obtained by applying step10 to the plurality of physical systems according to the order indicatedin the network structure of step 20.

An operation for transferring the energy state of step S10 in FIG. 2will be described in detail with reference to FIG. 3. FIG. 3 is aconceptual diagram of an operation (step S10 in FIG. 2) for transferringan energy state of one embodiment of the method of the presentinvention. First, the same quantum state indicated in the following (1)is prepared as an initial state in the two physical systems (u, d).|φ₀

φ₀|  (1)

At this point of time, the physical system u and the physical system dhave the same expectation value of energy indicated in the following(2).

φ₀ |Ĥ|φ ₀

  (2)

Next, dispersion of energy in a quantum state of (1) is given by thefollowing equation (3).ΔE=

φ ₀ |Ĥ ²|φ₀

−

φ₀ |Ĥ|φ ₀

²  (3)

As a “first operation 1”, time evolution of the quantum state of thephysical system u is performed for time τ (>0) using a Hamiltonianindicated in the following (4). A block denoted by a reference sign 1 inFIG. 3 indicates the first operation 1.Ĥ  (4)As a “second operation 2”, a quantum swapping operation is performedbetween the physical system u and the physical system d for time Δt(>0). A block denoted by a reference sign 2 in FIG. 3 indicates thesecond operation 2.

As a “third operation 3”, time evolution of the quantum state of thephysical system u is performed for time τ (>0) using an invertedHamiltonian indicated in the following (5). A block denoted by areference sign 3 in FIG. 3 indicates the third operation 3.−Ĥ  (5)

As a result of quantum mechanical time evolution defined by theoperations, in the quantum states of the physical system u and thephysical system d after the first to third operations in a series areperformed, the quantum state of the physical system u can be representedby the following equation (6), and the quantum state of the physicalsystem d can be represented by the following equation (7) in theapproximate range up to the first order of τ and Δt.ρ_(u)=|φ₀

φ₀|+τ[[Ĥ,|φ ₀

φ₀|],|φ₀

φ₀|]Δt  (6)ρ_(d)=|φ₀

φ₀|−τ[[Ĥ,|φ ₀

φ₀|],|φ₀

φ₀|]Δt  (7)

Here, in consideration of solution (9) of the following nonlinearequation (8),

$\begin{matrix}{{\frac{d}{dt}\left( {\left. \varphi_{t} \right\rangle\left\langle \varphi_{t} \right.} \right)} = {- {7\left\lbrack {\left\lbrack {\hat{H},{\left. \varphi_{t} \right\rangle\left\langle \varphi_{t} \right.}} \right\rbrack,{\left. \varphi_{t} \right\rangle\left\langle \varphi_{t} \right.}} \right\rbrack}}} & (8) \\{\left. \varphi_{t} \right\rangle\left\langle \varphi_{t} \right.} & (9)\end{matrix}$

It can be seen that the following equations (10) and (11) areestablished in the approximate range up to the first order of τ and Δt.ρ_(u)=|φ_(t=−Δt)

φ_(t=−Δt)|  (10)ρ_(d)=|φ_(t−|Δt)

φ_(t−|Δt)|  (11)

An expectation value of energy of the following (12) in a state given inthe above-described (9) is monotonously decreasing with respect to t.tr(Ĥ|φ _(t)

φ_(t)|)  (12)

An expectation value of energy in the state after the first to thirdoperations is represented by the following equation (13) with respect tothe physical system u, and is higher than an expectation value in theinitial state by ΔEτΔt.tr(Ĥρ _(u))=

φ₀ |Ĥ|φ

+τΔEΔt  (13)

On the other hand, the expectation value of energy is represented by thefollowing equation (14) with respect to the physical system d, and islower than the expectation value in the initial state by ΔEτΔt.tr(Ĥρ _(d))=

φ₀ |Ĥ|φ ₀

−τΔEΔt  (14)

As described above, it can be seen that the physical system u istransferred to the high energy state and the physical system d istransferred to the low energy state by the first to third series ofoperations.

Next, steps S20 and S30 in FIG. 2 will be described in detail withreference to FIGS. 4 to 6. In other words, a method used by the quantumcomputer for solving the discrete optimization problem by applying themethod of step S10 in FIG. 2 among the plurality of physical systemswill be described. FIG. 4 is a schematic diagram of the conceptualdiagram of FIG. 3. In FIG. 4, a black circle 4 and a white circle 5 aresimplified descriptions corresponding to the first operation 1 to thethird operation 3 in FIG. 3. FIG. 5 is a diagram illustrating step S20of constructing the network structure among the plurality of physicalsystems in one embodiment of the method of the present invention andstep S30 of obtaining the physical system having the minimum energystate among the plurality of physical systems. In FIG. 5, the states ofthe plurality of physical systems are displayed by using the simplifieddescriptions (black circle 4 and white circle 5) in FIG. 4. FIG. 6 is adiagram illustrating the time evolution of the expectation value ofenergy of a case (FIG. 5) in which the operation of step S10 is appliedto all the physical system pairs in the network structure illustrated inFIG. 5 by causing the two physical systems (u, d) to correspond thereto,and the physical system having the minimum energy state in the pluralityof physical systems is obtained.

m is set as a positive integer and 2m physical systems are considered. Anetwork structure among the plurality of physical systems is constructedby the following procedure. FIG. 5 illustrates a case of m=4 (physicalsystem 1 to physical system 8).

(1) A non-negative integer step starting from 0 is introduced. FIGS. 5and 6 illustrate a state transfer in which step=0 is set as an initialstate and step=1 to step=11 are sequentially increased by one.

(2) An integer s_(j) (step) determined by the following rules isintroduced into a physical system of j-th (j is a positive integer equalto or less than 2m−1).

(2-1) s_(j)(0) is set as 0 for all the physical systems j.

(2-2) A pair of a physical system j and a physical system j′ which havea state value s_(j) (step)=s_(j′) (step) is selected from all thephysical systems j and j′ where j<j′ (j′ is a positive integer equal toor less than 2m). In FIG. 5, one line that connects a black circle and awhite circle in a vertical direction (direction in which the physicalsystems 1 to 8 are arranged) represents one pair. For example, in thephysical systems 1 and 3 of step=2, s₁(2)=s₃(2)=−1, and the physicalsystem 1 and the physical system 3 are paired. In the same manner, inthe physical system 4 and the physical system 7 of step=7,s₄(7)=s₇(7)=+2, and the physical system 4 and the physical system 7 arepaired. Except for the physical system j and the physical system j′ oncepaired, such pairs are collected as much as possible.

(2-3) A set having a plurality of pair of the physical systemsconfigured as described above as an element is defined as Ω(step).

(2-4) With respect to the physical system j and physical system j′appearing in Ω(step), s_(j)(step+1) and s_(j′) (step+1) are defined asin the following equations (15) and (16).s _(j)(step+1=s _(j)(step)−1  (15)s _(j′)(step+1)=s _(j′)(step)+1  (16)

In FIG. 5, for example, in a pair of the physical system 3 and thephysical system 6 of step=5, s₃(6)=s₃(5)−1=−1, s₆(6)=s₆(5)+1=0. The sameapplies to the other pairs.

(2-5) With respect to j which does not appear in Ω(step), the followingequation (17) is defined.s _(j)(step+1)=s _(j)(step)  (17)

In FIG. 5, for example, from step=4 to step=7 of the physical system 1,s₁(7)=s₁(6)=s₁(5)=s₁(4)s₁(3)=−3. The same applies to others.

(2-6) Returning to the above-described (2-2) with step+1 as step, and(2-2) to (2-6) are repeated until the set Ω(step) becomes an empty set.

(2-7) A step obtained by subtracting 1 from a step where the set Ω(step) becomes the empty set is defined as step*. In FIG. 5, step=11obtained by subtracting 1 from step=12 (final state) where the setΩ(step) becomes the empty set becomes step*. At this time, the following(18) is referred to as a network structure among the plurality ofphysical systems. This network structure (18) can be determinedindependently of the Hamiltonian H which desires to obtain a minimumeigenstate.

$\begin{matrix}{\underset{{step}\mspace{14mu} 0}{\bigcup\limits^{{step}*}}{\Omega({step})}} & (18)\end{matrix}$

Further, the value of s_(j)(step*) in each physical system j can beobtained by the following equation (19) by a construction method for thenetwork structure among the plurality of physical systems.

$\begin{matrix}{{s_{j}({step})} = \left\{ \begin{matrix}{j - m - 1} & {for} & {j \in \left\{ {1,\ldots\;,m} \right\}} \\{j - m} & {for} & {j \in \left\{ {{m + 1},\ldots\;,{2m}} \right\}}\end{matrix} \right.} & (19)\end{matrix}$

In FIG. 5, for example, in the physical system 4,s₄(step*)=s₄(11)=j−m−1=4−4−1=−1, and in the physical system 8,s₈(step*)=s₈(11)=j−m=8−4=+4.

The time evolution of the quantum state is performed by using thenetwork structure among the plurality of physical systems constructed asdescribed above and the operation of transferring the energy state ofstep S10 in FIG. 2 (an energy exchange method between the two physicalsystems having the same quantum state) as follows.

(1) The same quantum state is prepared as an initial state in 2mphysical systems.

(2) Step=0 is set.

(3) The operation of transferring the energy state of step S10 isapplied to all the physical system pairs {j, j′} (note that j<j′)included in the set Ω(step) under the condition that the physical systemj corresponds to the physical system u of step S10 (FIG. 3) in FIG. 2,and the physical system j′ corresponds to the physical system d.

(4) Step+1 is returned to (3) as a new step, and repeated untilstep=step* is reached.

In this case, the above-described equations (10) and (11) are repeatedlyapplied, and the state of each step in each physical system j isrepresented by the following (20) in the approximate range up to thefirst order of τ and Δt by using solution (9) of equation (8).|φ_(t−s) _(j) _((step)Δt)

φ_(t−s) _(j) _((step)Δt)|  (20)

The result of strictly computing the expectation value of energy at thistime is shown in a diagram illustrating the time evolution of theexpectation value of energy in FIG. 6. In FIG. 6, it is shown that onearea surrounded by a broken line 7 represents the expectation value ofenergy; black and white shading represents the magnitude of theexpectation value of energy; and as the color becomes darker, theexpectation value of energy becomes smaller. Since FIG. 6 corresponds tothe physical system of m=4 in FIG. 5, it can be seen that theexpectation value of energy becomes minimum in step=11 of the physicalsystem 8.

In step=step*, equation (19), equation (20), and equation (12) aremonotonically decreasing with respect to t, whereby the expectationvalue of energy in the state of each physical system j is monotonouslydecreasing with respect to j. Among them, particularly, in the physicalsystem of j=2m (physical system 8 in the example of FIG. 5), the loweststate of the expectation value of energy of the following (21) isrealized as a result of the quantum mechanical time evolution defined bythe series of operations described above.|φ_(t−+mΔt)

ρ_(t−+mΔt)|  (21)

First Embodiment

As a first embodiment, an implementation example when the operation ofstep S10 in FIG. 2 is applied to a superconducting circuit usingJosephson junction (“CJJ-rf-SQUID”, Physical Review B 80,052506,2009)will be hereinafter described. FIG. 7 is a diagram illustrating theimplementation example (configuration example). FIG. 7 illustrates anexample of two quantum bits 1 and 2. In the physical system u and d, asuperconducting loop 10 forming the quantum bits 1 and 2 includes a miniloop 11 including two Josephson junctions (x). In each of thesuperconducting loops 10, the similar superconducting loops aremagnetically joined to each other through mutual inductance. Further, inthe following description, a case in which two quantum bits i and j aremore generally assumed will be described.

In the CJJ-rf-SQUID system, an interaction in the form of the followingequation (22) can be formed between the two quantum bits i and j.Ĥ _((i,j))=½[h _(i)(t){circumflex over (σ)}_(z) ^((i)) +h_(j)(t){circumflex over (σ)}_(z) ^((j))+Δ_(i)(t){circumflex over(σ)}_(x) ^((i))+Δ_(j)(t){circumflex over (σ)}_(x) ^((j)]+J)_(ij)(t){circumflex over (σ)}_(z) ^((i)){circumflex over (σ)}_(z)^((j))  (22)

Here, each coefficient of the following (23) in equation (22) is afunction of time which takes an integer value. FIG. 8 is a diagramillustrating the control sequence of the embodiment in FIG. 7, that is,the time transfer of values (−1, 0, +1) of each coefficient of thefollowing (23).h _(i)(t),h _(j)(t),Δ_(i)(t),Δ_(j)(t),J _(ij)(t)  (23)

Further, each coefficient of the following (24) is a Pauli matrix of thequantum bit i.{circumflex over (σ)}_(x) ^((i)){circumflex over (σ)}_(y)^((i)),{circumflex over (σ)}_(z) ^((i))  (24)

A set of quantum bits in which the interaction of the above-describedequation (22) is realized is considered. The set of these quantum bitsis divided into two, and one is caused to correspond to the physicalsystem u and the other is caused to correspond to the physical system d.At this time, a label of the quantum bit i belonging to the physicalsystem u is defined as (i, u), and a label of the quantum bit ibelonging to the physical system d is defined as (i, d). That is, theinteraction between these quantum bits is represented as the followingequation (25).

$\begin{matrix}{{\sum\limits_{q \in {\{{u,d}\}}}{\sum\limits_{i}{\frac{1}{2}\left( {{{h_{i,q}(i)}{\hat{\sigma}}_{z}^{({i,q})}} + {{\Delta_{i,q}(t)}{\hat{\sigma}}_{x}^{({i,q})}}} \right)}}} + {\sum\limits_{q \in {\{{u,d}\}}}{\sum\limits_{q^{\prime} \in {\{{u,d}\}}}{\sum\limits_{i,j}{J_{i,q,j,q^{\prime}}{\hat{\sigma}}_{z}^{({i,q})}{\hat{\sigma}}_{z}^{({j,q^{\prime}})}}}}}} & (25)\end{matrix}$

At this time, in the same manner as the case of the quantum annealingcomputation, in the interaction of equation (25) at t<0, the followingequation (26) is introduced and other coefficients are set to 0, therebypreparing an initial state (range of t<0 in FIG. 8).Δ_(i,u)(t)=Δ_(i,d)(t)=−1  (26)

In the same manner as the case of the quantum annealing computation,corresponding to the above-described first operation, the Hamiltonian ofequation (4) is realized as shown in the following equation (27) byusing the interaction between the quantum bits forming the physicalsystem u in equation (25).

$\begin{matrix}{\hat{H} = {{\sum\limits_{i}{\frac{1}{2}h_{i,u}{\hat{\sigma}}_{z}^{({i,u})}}} + {\sum\limits_{i,j}{J_{i,u,j,u}\sigma_{z}^{({i,u})}\sigma_{z}^{({j,u})}}}}} & (27)\end{matrix}$

That is, the following equations (28) and (29) are set for time τ (fortime τ in FIG. 8), thereby completing the first operation.h _(i,u)(t)=h _(i,u) ,J _(i,u,j,u)(t)=J _(i,u,j,u)  (28)h _(i,d)(t)=Δ_(i,u)(t)=Δ_(i,d)(t)=J _(i,u,j,u′)(t)=J_(i,u′,j,u′(t)=)0  (29)

Corresponding to the above-described second operation, a swappingoperation is performed. The swapping operation forms the interaction ofthe following equation (30) between the quantum bits forming thephysical system u and the physical system d in equation (25), and is setfor the time Δt, thereby completing the second operation.

$\begin{matrix}{{\hat{S}}_{ud} = {\frac{1}{2}\left( {I + {\sum\limits_{i}\left( {{{\hat{\sigma}}_{x}^{({i,u})}{\hat{\sigma}}_{x}^{({i,d})}} + {{\hat{\sigma}}_{y}^{({i,u})}{\hat{\sigma}}_{y}^{({i,d})}} + {{\hat{\sigma}}_{z}^{({i,u})}{\hat{\sigma}}_{z}^{({i,d})}}} \right)}} \right)}} & (30)\end{matrix}$

However, the interaction of equation (30) cannot be directly realized bythe interaction in the form of equation (25). Therefore, when thefollowing equations (31) to (34) are paid attention to, in the range upto the first order of Δt that establishes equation (6) and equation (7),the second operation can be implemented indirectly by performing thefollowing operations (1) to (3) in order.

$\begin{matrix}{e^{{- i}\;{\hat{S}}_{ud}\Delta\; t} = {{e^{{- i}\;\frac{\Delta\; t}{2}}U_{x}U_{y}U_{z}} + {O\left( {\Delta\; t^{2}} \right)}}} & (31) \\{U_{z} = {\exp\left( {{- i}\;\frac{\Delta\; t}{2}{\sum\limits_{i}{{\hat{\sigma}}_{z}^{({i,u})}{\hat{\sigma}}_{z}^{({i,d})}}}} \right)}} & (32) \\{U_{y} = {e^{{- i}\frac{\pi}{2}{\sum_{i}{({\sigma_{x}^{({i,u})} + \sigma_{x}^{({i,d})}})}}}U_{z}e^{{+ i}\frac{\pi}{2}{\sum_{i}{({\sigma_{x}^{({i,u})} + \sigma_{x}^{({i,d})}})}}}}} & (33) \\{U_{x} = {e^{{- i}\frac{\pi}{2}{\sum_{i}{({\sigma_{z}^{({i,u})} + \sigma_{z}^{({i,d})}})}}}U_{y}e^{{+ i}\frac{\pi}{2}{\sum_{i}{({\sigma_{z}^{({i,u})} + \sigma_{z}^{({i,d})}})}}}}} & (34)\end{matrix}$

(1) Operation of Equation (32):

In equation (25), as the following equation (35), all the remainingcoefficients are set to 0 for time Δt (for the first Δt in FIG. 8).Accordingly, the operation of equation (31) is completed.J _(i,u,i,d)(t)=1  (35)

(2) Operation of Equation (33):

In equation (25), as the following equation (36), all the remainingcoefficients are set to 0 for time π/2 (for the first π/2 in FIG. 8).After that, as equation (35), all the remaining coefficients are set to0 for time Δt (for the second Δt in FIG. 8).Δ_(i,u)(t)=Δ_(i,d)(t)=1  (36)

Further, as the following equation (37), all the remaining coefficientsare set to 0 for time π/2 (for the second π/2 in FIG. 8). Accordingly,the operation of equation (33) is completed.Δ_(i,u)(t)=Δ_(i,d)(t)=−1  (37)

(3) Operation of Equation (34):

In equation (25), as the following equation (38), all the remainingcoefficients are set to 0 for time π/2 (for the third π/2 in FIG. 8).h _(i,u)(t)=h _(i,d)(t)=1  (38)

Thereafter, as equation (36), all the remaining coefficients are set to0 and time π/2 is allowed to elapse (for the fourth π/2 in FIG. 8), andthen as equation (35), all the remaining coefficients are set to 0 fortime Δt (for the third Δt in FIG. 8). Further, as equation (37), all theremaining coefficients are set to 0 and time π/2 is allowed to elapse(for the fifth π/2 in FIG. 8), and further, as the following equation(39), all the remaining coefficients are set to 0 for time π/2 (for thelast π/2 in FIG. 8).h _(i,u)(t)=h _(i,d)(t)=−1  (39)

Accordingly, the operation of equation (34) is completed. The secondoperation is completed by the above-described series of operations (1),(2), and (3).

Corresponding to the above-described second operation, the interactionof equation (5) is set for time interval τ by using the interactionbetween the quantum bits forming the physical system u in equation (25).Since this is formed by the inversion of equation (27), the followingequation (40) may be set corresponding to equation (28). Accordingly,the third operation is completed.h _(i,d)(t)=−h _(i,u) ,J _(i,u,j,u)(t)=−J _(i,u,j,u)  (40)

As described above, in order to use the method (operation) of step S10in FIG. 2 by the implemented CJJ-rf-SQUID for the quantum computation bybeing repeatedly applied to 2m physical systems, the method can be used(implemented) by using the above-described series of processes withreference to FIGS. 4 to 6.

Second Embodiment

Next, as a second embodiment, in order to evaluate the number ofcomputation steps and the required number of quantum bits, a case inwhich the present invention is applied to a search problem of a searchspace 2^(n) will be described. First, the Hamiltonian of the followingequation (41) corresponding to the search problem is applied to thenonlinear equation of equation (8).Ĥ=−ω ₀|ω₀

ω₀|,ω₀>0  (41)

When the correct solution probability of the following equation (42) attime t is computed, an analytical solution exists and thus the followingequation (43) is obtained. FIG. 9 is a diagram illustrating n dependencyof a temporal behavior of a correct solution probability P(t) of thesearch problem based upon the nonlinear equation, obtained by plottingP(t) of equation (43).

$\begin{matrix}{{P(t)}:={\left\langle {\omega_{0}❘\varphi_{i}} \right\rangle }^{2}} & (42) \\{{P(t)} = \frac{1}{1 + {\left( {{P(0)}^{- 1} - 1} \right){\exp\left( {{- 2}\;\omega_{0}\tau\; t} \right)}}}} & (43)\end{matrix}$

Here, considered is a case where a size of a solution space is thefollowing (44) and an initial correct solution probability isrepresented by the following equation (45).2^(n)  (44)P(0)=2^(−n)  (45)

At this time, when time t satisfying the following equation (46) isdefined as t*, t* is obtained as the following equation (47). FIG. 10illustrates a plot of t* in equation (47), and a graph A shows ndependency of time t* when the correct solution probability P(t)≥0.9.

$\begin{matrix}{{P(t)} \geq \epsilon} & (46) \\{t^{*} = {{\frac{1}{2\;\omega_{0}\tau}\left( {{\ln\;\frac{1 - \epsilon}{\epsilon}} + {\ln\left( {2^{n} - 1} \right)}} \right)} = {O\left( \frac{n}{\omega_{0}\tau} \right)}}} & (47)\end{matrix}$

Here, according to a fact that the state obtained by this example isequation (21), when m corresponding to t* of equation (47) is set to m*,the following equation (48) can be obtained.

$\begin{matrix}{m^{*} = {{\frac{1}{2\omega_{0}\tau\;\Delta\; t}\left( {{\ln\frac{1 - \epsilon}{\epsilon}} + {\ln\left( {2^{n} - 1} \right)}} \right)} = {O\left( \frac{n}{\omega_{0}\tau\;\Delta\; t} \right)}}} & (48)\end{matrix}$

When noting that a relationship between step* and m is the followingequation (49) regardless of the details of the Hamiltonian (H), it canbe seen that the step* determined by m* is represented by the followingequation (50), which indicates the second order (n²) of n.

$\begin{matrix}{{step}^{*} = {O\left( m^{2} \right)}} & (49) \\{{step}^{*} = {O\left( \frac{n^{2}}{\omega_{0}^{2}{\tau\;}^{2}\Delta\; t^{2}} \right)}} & (50)\end{matrix}$

Further, since the number of quantum bits per system required forrepresenting the problem of equation (43) is n pieces, the followingequation (51) is obtained for the whole 2m* system.

$\begin{matrix}{{2m^{*}n} = {O\left( \frac{n^{2}}{\omega_{0}\tau\;\Delta\; t} \right)}} & (51)\end{matrix}$

Further, since the number C(m) of quantum operations required for thegiven m is the following equation (52), the following equation (53) isobtained when applied to the search problem. FIG. 11 is a diagramillustrating m dependency of the step* obtained by plotting equation(50) and the number C(m) of quantum operations obtained by plottingequation (53).

$\begin{matrix}{{C(m)} = {{\sum\limits_{{step} = 1}^{{step}^{*}}{{\Omega({step})}}} = {O\left( m^{3} \right)}}} & (52) \\{{C\left( m^{*} \right)} = {O\left( \frac{n^{3}}{\omega_{0}\tau\;\Delta\; t} \right)}} & (53)\end{matrix}$

When the conventional quantum annealing computation is applied to thesearch problem of the search space 2^(n) and the gate type quantumcomputation search problem, it is known that the number of computationsteps is 2^(n/2) (even though the number is smaller than that ofcomputation that does not use quantum) and behaves exponentially withrespect to n. On the other hand, as described above, the number ofcomputation steps (step*) according to the present invention issuppressed in a highly polynomial manner (for example, n²) with respectto n. In the same manner, the number of quantum bits required for thecomputation is proportional to n² in the present invention, whereas theconventional computation method is approximately proportional to n.Further, the problem can be solved by the number of computation stepsand the required number of quantum operations (C(m)) in a highlypolynomial size (for example, n³) with respect to n.

Embodiments of the present invention have been described with referenceto the accompanying drawings. However, the present invention is notlimited to the embodiments. Further, the present invention can beimplemented in a manner to which various improvements, corrections, andmodification are added based upon the knowledge of those skilled in theart without departing from the spirit of the present invention.

What is claimed is:
 1. A method for performing quantum annealingcomputation to solve a discrete optimization problem using a quantumannealing computer comprising the step of; (a) identifying an operationfor transferring energy state of one of two physical systems (u, d)having a same quantum state to a low energy state and transferringenergy state of the other of the two physical systems to a high energystate; (b) constructing a network structure among a plurality ofphysical systems that indicates an order of application of the operationof the step (a) on two physical systems among the plurality of physicalsystems; (c) obtaining a physical system having a minimum energy statein the plurality of physical systems by applying the operation of thestep (a) to the plurality of physical systems according to the orderindicated in the network structure of step (b).
 2. A method forperforming quantum annealing computation according to the claim 1,wherein the step (a) comprises the step of (a1) giving the same quantumstate as an initial state to the two physical systems (u, d); (a2)performing time evolving of the quantum state of the physical system ufor a time τ (>0) using a Hamiltonian (H); (a3) performing a quantumswapping operation between physical system u and physical system d fortime Δt (>0); and (a4) performing time evolving of the quantum state ofthe physical system u for time τ (>0) using an inverted Hamiltonian(−H).
 3. A method for performing quantum annealing computation accordingto the claim 2, wherein the step (c) comprises the step of (c1) giving asame quantum state as an initial state to 2m physical systems (m is apositive integer); (c2) setting step=0, and (c3) applying an operationincluding the steps (a1) to (a4) to all the physical system pairs (j,j′) in the set Ω (step), under the condition that the physical system jcorresponds to the physical system u and the physical system j′corresponds to the physical system d.
 4. A method for performing quantumannealing computation according to the claim 1, wherein the step (b)comprises the step of (b1) defining a state value of j-th physicalsystem j (j is a positive integer equal to or less than 2m−1 and m is apositive integer) as j (step) in 2m physical systems and setting s j(0)=0 for all physical systems j; (b2) selecting a plurality of pairs ofthe physical system j and the physical system j′ which have a statevalue sj (step)=s j′ (step), from all the physical systems j and j′where j<j′ (j, j′ is a positive integer of 2 m or less); (b3) collectingthe plurality of pairs of the physical system j and the physical systemj′ as a set Ω (step); (b4) setting the physical system j and physicalsystem j′ included in the set Ω (step) to s j (step+1)=s j (step)−1,s j′(step+1)=s j′ (step)+1; (b5) setting the physical system j not includedin the set Ω (step) to s j (step+1)=s j (step) and step+1 to step; (b6)repeating the steps (b2) to (b5) until the set Ω (step) becomes an emptyset; and (b7) setting a step obtained by subtracting 1 from the step inwhich the set Ω (step) is an empty set, to a step *.
 5. A method forperforming quantum annealing computation according to the claim 4,wherein the state value s j (step *) in each of the physical system j isgiven as s j (step *)=j−m−1 (j=1, 2, . . . , m) or s j (step *)=j−m(j=m+1, m+2, . . . , 2m).